Mathematical Programming

, Volume 157, Issue 1, pp 153–189

Chance-constrained problems and rare events: an importance sampling approach

  • Javiera Barrera
  • Tito Homem-de-Mello
  • Eduardo Moreno
  • Bernardo K. Pagnoncelli
  • Gianpiero Canessa
Full Length Paper Series B
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Abstract

We study chance-constrained problems in which the constraints involve the probability of a rare event. We discuss the relevance of such problems and show that the existing sampling-based algorithms cannot be applied directly in this case, since they require an impractical number of samples to yield reasonable solutions. We argue that importance sampling (IS) techniques, combined with a Sample Average Approximation (SAA) approach, can be effectively used in such situations, provided that variance can be reduced uniformly with respect to the decision variables. We give sufficient conditions to obtain such uniform variance reduction, and prove asymptotic convergence of the combined SAA-IS approach. As it often happens with IS techniques, the practical performance of the proposed approach relies on exploiting the structure of the problem under study; in our case, we work with a telecommunications problem with Bernoulli input distributions, and show how variance can be reduced uniformly over a suitable approximation of the feasibility set by choosing proper parameters for the IS distributions. Although some of the results are specific to this problem, we are able to draw general insights that can be useful for other classes of problems. We present numerical results to illustrate our findings.

Keywords

Chance-constrained programming Sample average approximation Importance sampling Rare-event simulation 

Mathematics Subject Classification

90C15 (Mathematical Programming/Stochastic programming) 65C05 (Numerical Analysis/Probabilistic Methods/Monte Carlo methods) 68M10 (Computer Science/Network design and communication) 

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Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2015

Authors and Affiliations

  1. 1.Faculty of Engineering and SciencesUniversidad Adolfo IbáñezSantiagoChile
  2. 2.School of BusinessUniversidad Adolfo IbáñezSantiagoChile
  3. 3.IE&OR ProgramUniversidad Adolfo IbáñezSantiagoChile

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